It is seen that these rogue waves comprise a number of Peregrine waves forming triangular patterns for large \(a_3\) , pentagon patterns for large \(a_5\) , heptagon patterns for large \(a_7\) , nonagon patterns for large \(a_9\) , hendecagon (eleven-sided polygon) patterns for large \(a_{11}\) , and tridecagon (thirteen-sided polygon) patterns for large \(a_{13}\) . In the literature, patterns on the diagonal of Fig. REF  are sometimes called single-shell ring structures [1]}. In addition to these Peregrine waves away from the origin, some of the rogue waves also contain a lower-order rogue wave at their centers. For instance, for the 7-th order rogue waves in the bottom row of Fig. REF , the first and fourth panels (with large \(a_3\)  and \(a_9\)  respectively) exhibit a Peregrine wave in their centers; the second panel (with large \(a_5\) ) exhibits a second-order rogue wave in the center; the fifth panel (with large \(a_{11}\) ) exhibits a third-order rogue wave in the center; and the last panel (with large \(a_{13}\) ) exhibits a fifth-order rogue wave in the center. For our choices of parameters in rogue waves of Fig. REF , these lower-order rogue waves in the center are all super-rogue waves, i.e., rogue waves with the highest peak amplitude of their orders. We note by passing that the first five rows of rogue patterns in Fig. REF  resemble those plotted in Ref. [2]} from Akhmediev breathers in the rogue-wave limit, although orientations between the two sets of patterns are very different.
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